directed object' (Rev #6, changes)

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Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$. Then the cospan category (same objects and cospans as morphisms) is $V$-enriched, too.

An *interval object* is defined to be a cospan $*\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*$.

The pushout $I^{\coprod_2}$ of this diagram satisfies ${}_*[ I,I^{\coprod_2}]_*\simeq$ is contractible (see co-span for this notation).

Let $V$ be a monoidal category, let $C$ be $V$-enriched, closed monoidal homotopical category, let $*$ denote the tensor unit of $C$ which we assume to be the terminal object, let $*\stackrel{0}{\to} I\stackrel{1}{\leftarrow}$ denote the interval object of $C$, let $X$ be a pointed object of $C$. Let $D$ be the~~ functor~~ bifunctor$D:C\times C\to V$~~ D:C\to~~ D:C\times C\to V, $(X,I)\mapsto [I,X]$~~ X\mapsto~~ (X,I)\mapsto [I,X].

A *direction in $C$* is defined to be a subfunctor $d$ of $D(I,-)$~~ D~~ D(I,-) preserving the terminal object and satisfying the properties below. In this case $d X$ is called a *direction for $X$*. A global element of $dX$ is called a *$d$-directed path in $X$* and their collection we denote by $ddp(X)$~~ ~~ .~~ the~~ The properties are:

(1) The $D$ image of every map $I\to *\to X$ factoring over the point is in $ddp(X)$.

(2) $\mathrm{ddpf}(,\mathrm{Xg})\in \mathrm{dX}$~~ ddp(X)~~ f,g\in dX ~~ is~~ are~~ closed~~ called~~ under~~ to~~ the~~ be~~ tensor~~ a~~ product.~~~~ For~~~~$\alpha,\beta\in ddp(X)$~~*composable pair* ,~~ their~~ if~~ pushout~~$\alpha \overline{f}\otimes \circ \beta 0=\overline{g}\circ 1$~~ \alpha\otimes~~ \overline~~ \beta~~ f\circ 0=\overline g\circ 1 where the overlined letters denote the adjoints under the hom adjunction. The composition of a composable pair is~~ called~~ defined~~ their~~ by$f\circ g:=\underline{\overline{f}\bullet\overline{g}}$ . Then the condition is that$ddp(X)$ is closed under composition.*composition*

~~ A~~ (3) If$\tau\in hom (I,I)$ *directed object*~~ is~~ and~~ defined~~~~ to~~~~ be~~~~ a~~~~ pair~~${}_{d}fX\in :\mathrm{ddp}=(X,\mathrm{dX})$~~ {}_d~~ f\in~~ X:=(X,dX)~~ ddp(X) ~~ consisting~~ then~~ of~~~~ an~~~~ object~~$X\underline{\overline{f}\circ \tau}\in \mathrm{dX}$~~ X~~ \underline{\overline{f}\circ \tau}\in dX ~~ of~~ where the underlined term denotes the adjoint (in the other direction) under the hom adjunction.~~$C$~~~~ and a direction ~~~~$dX$~~~~ for ~~~~$X$~~~~.~~

~~ A~~ (4)$ddp(X)$ is a functor*morphism of directed objects*~~$f:(X,dX)\to (Y,dy)$~~~~ is defined to be a pair ~~~~$(f,df)$~~~~ making~~

$\array{
X&
\stackrel{f}{\to}&
Y
\\
\downarrow^d&&\downarrow^d
\\
dX&
\stackrel{df}{\to}&
d Y
}$

A *directed object* is defined to be a pair ${}_d X:=(X,dX)$ consisting of an object $X$ of $C$ and a direction $dX$ for $X$.

A *morphism of directed objects* $f:(X,dX)\to (Y,dY)$ is defined to be a pair $(f,df)$ making

Let $C$ be $V$-enriched, closed monoidal homotopical category, let $I$ denote the tensor unit of $C$.

A *directed-path-space objects* is defined.